A MACRODISPERSION MODEL FOR PLUME MIGRATION IN WEAKLY HETEROGENEOUS MEDIA (ln K VARIANCE 0.5) WITH PORESCALE DISPERSION  
Parameters & Legend  Problem Statement  Key Observations  Additional Observations  Mathematical Interpretation  




A MACRODISPERSION MODEL FOR PLUME MIGRATION IN WEAKLY HETEROGENEOUS MEDIA (ln K VARIANCE 0.5) WITH PORESCALE DISPERSION 

Problem Statement  
This video compares two different modeling techniques to predict macrodispersion of a conservative solute plume in a weakly heterogeneous medium. The first technique employs a detailed description of spatial variability of conductivity using statistical parameters, and a porescale dispersion coefficient. The second technique uses Gelhar's asymptotic macrodispersion model ( Gelhar and Axness, 1983a ) that consists of a homogeneous medium and a macrodispersion coefficient. This video demonstrates the effectiveness of Gelhar's model in predicting plume spreading. The modeling domain consists of constant head boundaries on the left and right extremes, and noflow boundaries at the top and bottom; the initial size of the plume is significantly larger than the correlation scale of heterogeneity. Details are provided in Table 3.1. 

Key Observations  
The following observations can be made from the video:
Similar experiments with larger perturbations (higher ln K variance) are shown in subsequent videos. 

Additional Observations  
Gelhar's macrodispersion model attempts to predict the largescale spreading that occurs in a heterogeneous field. The underlying assumption for the macrodispersion model is that the conductivity perturbations in the heterogeneous field are small. These perturbations control plume spreading in the heterogeneous field. Longitudinal spreading in the heterogeneous field is greater than transverse spreading. The macrodispersion model predicts that the longitudinal spreading is directly proportional to the correlation scale of heterogeneity and the ln K variance, and transverse spreading is directly proportional to the porescale dispersivity and the ln K variance. Since the correlation scale is at least two orders of magnitudes larger than porescale dispersivity, longitudinal spreading is significantly larger than its transverse counterpart. Therefore, the macrodispersion model is able to predict the overall spreading of the plume approximately. Since the macrodisperion model consists of a homogeneous medium, the plume obtained is always regular. As a result, the irregular shape of the plume seen in the heterogeneous field is not predicted. The irregular shape of the plume translates to irregular variations of concentration, which are also not predicted. Consequently, the macrodispersion model is unable to accurately predict the maximum concentration and the degree of dilution of the plume. 

Mathematical Interpretation  
The following equations show the relationship between porescale dispersion and macrodispersion:
It can be shown [ Gelhar and Axness, 1983a ] that:
The coefficient A_{ii}is called macrodispersivity and is given by [ Gelhar, 1993 ]:
where: = ln K variance; λ = Correlation scale; A_{11} = Longitudinal macrodispersivity; A_{22} = Transverse macrodispersivity; α_{L}= Longitudinal dispersivity; α_{T}= Transverse dispersivity. Equation (3.1.3b) shows that the longitudinal macrodispersion is unaffected by porescale dispersion, whereas transverse macrodispersion is affected. Therefore, the magnitude of longitudinal macrodispersion is many orders of magnitude larger than the transverse macrodispersion. Substituting (3.1.3a) into (3.1.2):
The porescale dispersivity, α_{ij}, is typically onehundredth of a meter, while the macrodispersivity, A_{ij} , is approximately one meter. Therefore, macrodispersion is the dominant process and controls the total spreading of the plume. Equation (3.1.5) simplifies the complex heterogeneous field, represented by (3.1.2). A simple mean macrodispersion model is obtained by replacing the dispersion coefficient with a macrodispersivity value. This model can predict the mean spreading of the plume. 